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Unformatted text preview: t size took a Cray computer just a few hours [440]. In 1988, Carl Pomerance designed a modular factoring machine, using custom VLSI chips [1259]. The size of the number you would be able to factor depends on how large a machine you can afford to build. He never built it. In 1993, a 120digit hard number was factored using the quadratic sieve; the calculation took 825 mipsyears and was completed in three months real time [463]. Other results are [504]. Today’s factoring attempts use computer networks [302, 955]. In factoring a 116digit number, Arjen Lenstra and Mark Manasse used 400 mipsyears—the spare time on an array of computers around the world for a few months. In March 1994, a 129digit (428bit) number was factored using the double large prime variation of the multiple polynomial QS [66] by a team of mathematicians led by Lenstra. Volunteers on the Internet carried out the computation: 600 people and 1600 machines over the course of eight months, probably the largest ad hoc multiprocessor ever assembled. The calculation was the equivalent of 4000 to 6000 mipsyears. The machines communicated via electronic mail, sending their individual results to a central repository where the final steps of analysis took place. This computation used the QS and fiveyearold theory; it would have taken onetenth the time using the NFS [949]. According to [66]: “We conclude that commonly used 512bit RSA moduli are vulnerable to any organization prepared to spend a few million dollars and to wait a few months.” They estimate that factoring a 512bit number would be 100 times harder using the same technology, and only 10 times harder using the NFS and current technology [949]. To keep up on the state of the art of factoring, RSA Data Security, Inc. set up the RSA Factoring Challenge in March 1991 [532]. The challenge consists of a list of hard numbers, each the product of two primes of roughly equal size. Each prime was chosen to be congruent to 2 modulo 3. There are 42 numbers in the challenge, one ea...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
 Fall '10
 ALIULGER
 Cryptography

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